Optimal. Leaf size=847 \[ \frac{2 \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{693 c^2 e^5}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-4 c^3 e (64 b d-69 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (41 b^2 d^2-92 a b e d+60 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]
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Rubi [A] time = 2.70071, antiderivative size = 847, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {734, 814, 843, 718, 424, 419} \[ \frac{2 \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{693 c^2 e^5}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-4 c^3 e (64 b d-69 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (41 b^2 d^2-92 a b e d+60 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]
Antiderivative was successfully verified.
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Rule 734
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{5 \int \frac{(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 e}\\ &=\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}+\frac{10 \int \frac{\left (\frac{1}{2} \left (9 c e (b d-2 a e)^2-2 (2 c d-b e) \left (b d \left (4 c d-\frac{3 b e}{2}\right )-a e \left (c d+\frac{b e}{2}\right )\right )\right )-2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{231 c e^3}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{4 \int \frac{\frac{1}{4} \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (8 b c d^2-3 b^2 d e-2 a c d e-a b e^2\right )\right )\right )+\frac{1}{4} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{693 c^2 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{\left ((2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{693 c^2 e^6}-\frac{\left (4 \left (-\frac{1}{4} d (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right )+\frac{1}{4} e \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (8 b c d^2-3 b^2 d e-2 a c d e-a b e^2\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{693 c^2 e^6}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}-\frac{\left (8 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{1}{4} d (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right )+\frac{1}{4} e \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (8 b c d^2-3 b^2 d e-2 a c d e-a b e^2\right )\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+276 a c^3 d^2 e^2+5 b^3 c d e^3-276 a b c^2 d e^3+2 b^4 e^4-21 a b^2 c e^4+180 a^2 c^2 e^4\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.9757, size = 10879, normalized size = 12.84 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.401, size = 12152, normalized size = 14.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{\sqrt{e x + d}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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